Convexity Of A Zero-Coupon Bond

Why a Zero-Coupon Bond Has a Simple Sensitivity Profile

A zero-coupon bond makes one payment at maturity and no coupon payments before then. Its price therefore depends on a single discounted cash flow. This structure makes it possible to see directly how the payment date determines the bond’s duration and convexity, and how those measures can be used to estimate the price effect of a yield change.

The calculations below assume a fixed redemption payment, no default, annual compounding, and a positive annual effective yield $y$. The yield change is measured in annual-rate units: for example, 25 basis points is $0.0025$, not $25$ or $0.25$. The duration-convexity approximation is local; it estimates a price change near the current yield rather than calculating a new price exactly.

Definition: For a zero-coupon bond with redemption payment $F$ at time $n$, its price at annual effective yield $y$ is $P=F(1+y)^{-n}$. Its Macaulay duration is $n$ years, its modified duration is $D_{\mathrm{mod}}=n/(1+y)$, and its convexity with respect to the annual effective yield is $C=P''(y)/P(y)=n(n+1)/(1+y)^2$. These definitions assume a known, fixed payment date and amount and annual compounding.

Macaulay duration describes the timing of the bond’s cash flows, using present-value weights. Since the only cash flow occurs at time $n$, all of the present-value weight belongs to that date. Modified duration describes the first-order percentage price sensitivity to yield; convexity describes the second-order curvature of price against yield.

\frac{\Delta P}{P}\approx -D_{\mathrm{mod}}\Delta y+\frac{1}{2}C(\Delta y)^2

Here $\Delta y$ is the change in annual effective yield in decimal form. A positive yield change usually lowers price, represented by the negative duration term. For a positive-payment zero-coupon bond, convexity is positive, so the second-order term is positive whether yields rise or fall. The formula estimates the proportional change; multiplying by the current price converts that estimate into dollars.

Why the Duration and Convexity Formulas Hold

The formulas follow by differentiating the price function. This short derivation also shows why maturity, rather than a weighted average of several payment dates, determines the duration of this bond.

P(y)=F(1+y)^{-n}

Differentiating once gives the sensitivity of price to yield. Dividing the derivative by price gives the proportional sensitivity:

P'(y)=-nF(1+y)^{-n-1} \qquad\Longrightarrow\qquad \frac{P'(y)}{P(y)}=-\frac{n}{1+y}

Thus modified duration is the negative of the proportional first derivative, $D_{\mathrm{mod}}=-P'(y)/P(y)=n/(1+y)$. Macaulay duration is $n$, because the one and only payment occurs at time $n$; under annual compounding, modified duration is Macaulay duration divided by $1+y$.

Differentiating a second time and dividing by the original price gives convexity:

P''(y)=n(n+1)F(1+y)^{-n-2} \qquad\Longrightarrow\qquad \frac{P''(y)}{P(y)}=\frac{n(n+1)}{(1+y)^2}

A second-order Taylor approximation of price around the current yield is $P(y+\Delta y)\approx P(y)+P'(y)\Delta y+\tfrac12P''(y)(\Delta y)^2$. Dividing by $P(y)$ produces the duration-convexity formula. The approximation omits third- and higher-order terms, so it is not an exact repricing method, especially for a large yield shock.

Identify the payment date and yield convention.
For an annual-compounding zero-coupon bond, record maturity $n$ in years and annual effective yield $y$.

Calculate the current price and sensitivities.
Use $P=F(1+y)^{-n}$, $D_{\mathrm{mod}}=n/(1+y)$, and $C=n(n+1)/(1+y)^2$.

Convert the shock and estimate the change.
Express the yield change as a decimal, substitute it for $\Delta y$, and multiply the resulting proportional change by $P$ for a dollar estimate.

Worked Calculations

Worked Example: A Yield Increase for a Five-Year Zero-Coupon Bond

A hypothetical bond pays $1,000 in five years and has an annual effective yield of 4.00%. Estimate its price change if the yield increases by 50 basis points. Report the estimated new price to the nearest cent. Use the duration-convexity approximation.

Method. Calculate the current price, modified duration, and convexity, then apply the second-order approximation. This method avoids treating the yield shock as a percentage of the current yield.

The current price is

P=\frac{1000}{(1.04)^5} =\frac{1000}{1.2166529024} \approx \$821.9271

The Macaulay duration is the payment time, 5 years. Modified duration and convexity are

D_{\mathrm{mod}}=\frac{5}{1.04}\approx 4.8076923 \qquad C=\frac{5(6)}{(1.04)^2} =\frac{30}{1.0816} \approx 27.7366864

The shock is 50 basis points, or $\Delta y=0.005$. The estimated proportional change is

\frac{\Delta P}{P} \approx -(4.8076923)(0.005) +\frac12(27.7366864)(0.005)^2 \approx -0.0236917529

This is an estimated decrease of about 2.3692%. The dollar change is

\Delta P\approx (-0.0236917529)(\$821.9271) \approx -\$19.4729

The estimated new price is $\$821.9271-\$19.4729\approx \$802.4542$, or $802.45 to the nearest cent. The result is a decrease, as expected when yield rises. The convexity term offsets part of the duration-only decrease, as it should for positive convexity.

Common mistake avoided: Using $0.50$ as the yield change would mean a 50-percentage-point shock, not a 50-basis-point shock. The correct decimal change is $50/10{,}000=0.005$.

Worked Example: A Yield Decrease for a Three-Year Bond

A hypothetical zero-coupon bond pays $1,000 in three years. Its annual effective yield is 5.00%. Estimate the price increase when the yield falls by 100 basis points. Give the estimated dollar change and new price to the nearest cent.

Method. Use the same price derivatives, but retain the negative sign of the yield change. This avoids treating a yield decrease as if it were an increase in the duration term.

First calculate the price and sensitivities:

P=\frac{1000}{(1.05)^3} =\frac{1000}{1.157625} \approx \$863.8376

D_{\mathrm{mod}}=\frac{3}{1.05}\approx 2.8571429 \qquad C=\frac{3(4)}{(1.05)^2} =\frac{12}{1.1025} \approx 10.8843537

A decrease of 100 basis points means $\Delta y=-0.01$. Therefore,

\frac{\Delta P}{P} \approx -(2.8571429)(-0.01) +\frac12(10.8843537)(-0.01)^2 \approx 0.0291156463

The estimated price increase in dollars is

\Delta P\approx (0.0291156463)(\$863.8376) \approx \$25.1512

The estimated new price is $\$863.8376+\$25.1512\approx \$888.9888$, or $888.99 to the nearest cent. Both the duration and convexity contributions are positive for a yield decrease, so an increase in price is consistent with the model.

Common mistake avoided: The convexity term uses $(\Delta y)^2$, which is positive for either direction of shock. Do not make the convexity contribution negative just because the yield falls.

Worked Example: Estimating a Dollar Change from a Known Current Price

A hypothetical four-year zero-coupon bond currently costs $1,000 and has an annual effective yield of 5.00%. Estimate its price after a yield increase of 100 basis points. Report the dollar change and estimated new price to the nearest cent.

Method. The current price is supplied, so use it directly when converting the proportional change into dollars. This avoids assuming that the redemption payment is $1,000 merely because the current price is $1,000.

The bond’s Macaulay duration is 4 years. Its modified duration and convexity are

D_{\mathrm{mod}}=\frac{4}{1.05}\approx 3.8095238 \qquad C=\frac{4(5)}{(1.05)^2} =\frac{20}{1.1025} \approx 18.1405896

The 100-basis-point increase is $\Delta y=0.01$. The estimated proportional price change is

\frac{\Delta P}{P} \approx -(3.8095238)(0.01) +\frac12(18.1405896)(0.01)^2 \approx -0.0371882086

Using the supplied current price, the dollar change is

\Delta P\approx (-0.0371882086)(\$1{,}000) \approx -\$37.1882

The estimated new price is $\$1{,}000-\$37.1882\approx \$962.8118$, or $962.81 to the nearest cent. The estimate is a decrease because the first-order duration effect is negative and larger in magnitude than the positive convexity adjustment.

Common mistake avoided: Current price and redemption payment are different quantities in general. The redemption amount is not needed here because the current price is already given and the proportional sensitivity depends on $n$ and $y$.

Reading the Sensitivity Measures Correctly

For a zero-coupon bond, Macaulay duration equals maturity because there is no earlier cash flow to affect the present-value-weighted payment time. This does not mean that a bond’s percentage price sensitivity is simply its maturity. Under annual compounding, the first-order sensitivity is modified duration, $n/(1+y)$, which adjusts for the yield convention.

The convexity formula also depends only on maturity and yield for this bond. As maturity increases, the factors $n$ and $n+1$ increase; as yield changes, the denominator $(1+y)^2$ changes. Convexity is positive when the only cash flow is a positive redemption payment. This conclusion relies on the fixed, positive payment: it is not a general claim about instruments with signed cash flows or changing cash flows.

These formulas are specific to annual compounding and a yield quoted as an annual effective rate. If a problem instead specifies a nominal yield with a different compounding frequency, the price function and derivatives must use that convention consistently. Do not insert a nominal yield directly into the annual-effective formulas without adjustment.

Quantity	Zero-coupon bond result	Interpretation
Macaulay duration	$n$	The single payment occurs at maturity.
Modified duration	$n/(1+y)$	First-order proportional price sensitivity to annual effective yield.
Convexity	$n(n+1)/(1+y)^2$	Second derivative of price, divided by price.
Approximate proportional price change	$-D_{\mathrm{mod}}\Delta y+\tfrac12C(\Delta y)^2$	Second-order estimate for a yield shock in decimal form.

A practical check is to examine the signs before calculating dollars. With a positive yield and a positive redemption payment, a small yield increase should produce a negative duration effect and a positive convexity correction. A small yield decrease should produce two positive contributions. A result that reverses these signs usually indicates an error in the shock, the duration term, or the yield units.

Key takeaway: A zero-coupon bond has Macaulay duration equal to its payment date because all present value is concentrated at maturity. For annual effective yield $y$, use $D_{\mathrm{mod}}=n/(1+y)$ and $C=n(n+1)/(1+y)^2$, then apply the duration-convexity approximation with $\Delta y$ in decimal form.

Check Your Understanding

For each question, use annual compounding and the duration-convexity approximation where requested. Treat yield changes as decimal changes.

A zero-coupon bond pays $1,210 in two years and yields 10.00%. Calculate its current price, modified duration, and convexity. (Answer: Price $1,000.00; modified duration 1.8182 years; convexity 4.9587 years squared.)
A six-year zero-coupon bond has an annual effective yield of 4.00%. Calculate its Macaulay duration and modified duration. (Answer: Macaulay duration 6 years; modified duration $6/1.04\approx 5.7692$ years.)
A three-year zero-coupon bond pays $2,000 and yields 5.00%. Estimate the proportional price change if its yield decreases by 50 basis points. (Answer: Approximately $+1.4422\%$.)
A two-year zero-coupon bond pays $1,000 and yields 5.00%. Estimate the proportional price change for a yield increase of 100 basis points. (Answer: Approximately $-1.8776\%$.)
A four-year zero-coupon bond has an annual effective yield of 6.00%. By what factor does its convexity change if the yield rises to 8.00%, with maturity unchanged? (Answer: The convexity is multiplied by $(1.06/1.08)^2\approx 0.9633$, a decrease of approximately 3.67%.)

Quantity	Zero-coupon bond result	Interpretation
Macaulay duration	\(n\)	The single payment occurs at maturity.
Modified duration	\(n/(1+y)\)	First-order proportional price sensitivity to annual effective yield.
Convexity	\(n(n+1)/(1+y)^2\)	Second derivative of price, divided by price.
Approximate proportional price change	\(-D_{\mathrm{mod}}\Delta y+\tfrac12C(\Delta y)^2\)	Second-order estimate for a yield shock in decimal form.

What You'll Learn